Dual frequency density meter

ABSTRACT

An apparatus for measuring density of media moving through a conduit included therein comprises one or two vibrating sections under two different flexural vibrations vibrating at two different natural frequencies which are functions of the stiffness of the vibrating sections of the conduit, density of the media moving therethrough and viscosity of the media as well as the viscosity of the ambient air surrounding the vibrating sections of the conduit. A mathematical combination of the two different natural frequencies of the vibrating section or sections of the conduit eliminates the dependence thereof on the viscosities and determines the density of media accurately independent of the viscosity.

BACKGROUND OF THE INVENTION

An apparatus measuring density of media comprises a single section ofconduit under flexural vibrations in two lateral directions at twodifferent natural frequencies or two sections of conduit under flexuralvibrations at two different natural frequencies, wherein the density ofthe media moving through the conduit is determined from a combination ofthe two different natural frequencies, in which combination the effectof the viscosity of the media on the natural frequencies is eliminatedand, consequently, the density of the media is determined accuratelywithout errors introduced by the effect of the viscosity.

There are many instances in industrial processes and controls handlingthe flow fluids wherein the density of the moving fluid has to bemeasured accurately. One particular application of density measurementis to determine the mass flow rate of a fluid medium as a product of thefluid density measured by a density meter and a volume flow rate of thefluid measured by a volumetric flowmeter. There are mass flowmetersavailable at the present time such as the Coriolis force or convectiveinertia force mass flowmeters and thermal probe mass flowmeters. Thesetypes of mass flowmeters work poorly in measuring flows of highlyviscous fluids due to the error in the data acquisition yielding themass flow rate arising from the effect of the viscosity of the fluid,while they function excellently in the mass flow measurement of lowviscosity fluids. One of the more promising approaches to measurement ofthe mass flow rate is to employ a combination of an accurate densitymeter and a reliable positive displacement volumetric flowmeter, whichcombination is particularly effective in measuring mass flow rates ofhighly viscous fluids or mixtures of gaseous and fluid medium.

SUMMARY OF THE INVENTION

The primary object of the present invention is to provide a densitymeter employing one or two sections of a conduit vibrating at differentnatural frequencies, wherein the density of the fluid moving through thevibrating conduit is determined from a combination of the two differentnatural frequencies

Another object is to provide a density meter employing one or twosections of a conduit under flexural vibrations, wherein the density ofthe fluid moving through the conduit is determined independent of theviscosity of the fluid.

A further object is to provide a density meter comprising a singlesection of a conduit under flexural vibrations in two orthogonal lateraldirections at two different natural frequencies.

Yet another object is to provide a density meter comprising two sectionsof a conduit under flexural vibrations of two different naturalfrequencies, which two sections of the conduit are disposed in series.

Yet a further object is to provide a density meter comprising twosections of a conduit under flexural vibrations at two different naturalfrequencies, which two sections of the conduit are disposed in parallelarrangement.

Still another object is to provide a density meter including one or twosections of a conduit under continuously induced flexural vibrations attwo different natural frequencies.

Still a further object is to provide a density meter including one ortwo sections of a conduit under intermittently induced flexuralvibrations at two different natural frequencies.

These and other objects of the present invention -will become clear asthe description thereof progresses.

BRIEF DESCRIPTION OF THE FIGURES

The present invention may be described with a great clarity andspecificity by referring to the following figures :

FIG. 1 illustrates an embodiment of the dual frequency density meter ofthe present invention including two vibrating sections of a conduitdisposed in series, wherein the conduit includes a 360 degree loop.

FIG. 2 illustrates a cross section of the first half of the loopingconduit vibrating at a first natural frequency.

FIG. 3 illustrates a cross section of the second half of the loopingconduit vibrating at a second natural frequency.

FIG. 4 illustrates an embodiment of the dual frequency density meterincluding two U-shaped vibrating sections of a conduit disposed inseries.

FIG. 5 illustrates a cross section of the first U-shaped section of theconduit.

FIG. 6 illustrates a cross section of the second U-shaped section of theconduit.

meter

FIG. 7 illustrates an embodiment of the dual frequency density includingtwo straight vibrating sections of a conduit disposed in an U-shapedarrangement.

FIG. 8 illustrates an embodiment of the dual frequency density metercomprising a pair of angled vibrating sections of a conduit disposed inan U-shaped arrangement.

FIG. 9 illustrates an embodiment of the dual frequency density meterincluding a pair of U-shaped vibrating sections of a conduit disposed ina looping arrangement.

FIG. 10 illustrates an embodiment of the dual frequency density meterincluding an over-hanging vibrating section of a conduit.

FIG. 11 illustrates an embodiment of the dual frequency density meterincluding a pair of U-shaped vibrating conduits included in a 360 degreeloop of a conduit.

FIG. 12 illustrates an embodiment of the dual frequency density meterincluding a single straight vibrating section of a conduit.

FIG. 13 illustrates a cross section of the straight vibrating section ofthe conduit included in the embodiment shown in FIG. 12.

FIG. 14 illustrates an embodiment of the dual frequency density metercomprising a single straight vibrating section of a conduit with oneextremity including a flexible coupling.

FIG. 15 illustrates a cross section of the straight vibrating section ofthe conduit included in the embodiment shown in FIG. 14.

FIG. 16 illustrates another embodiment of the dual frequency densitymeter comprising a vibrating section of a conduit with one extremityincluding a flexible coupling.

FIG. 17 illustrates a further embodiment of the dual frequency densitymeter comprising a vibrating section of a conduit with one extremityincluding a flexible coupling.

FIG. 18 illustrates an embodiment of the dual frequency density metercomprising a straight vibrating section of a conduit with bothextremities respectively including two flexible couplings and twoorthogonal flexural motion restrainers.

FIG. 19 illustrates an embodiment of the dual frequency density metercomprising a straight vibrating section of a conduit with a flexiblecoupling included at a midsection thereof and a pair of orthogonalflexural motion restrainers respectively disposed at two opposite sidesof the flexible coupling.

FIG. 20 illustrates an embodiment of the dual frequency density meterincluding two straight vibrating sections of a conduit respectivelyincluding two flexible couplings, which vibrating sections are disposedin series.

FIG. 21 illustrates an embodiment of the dual frequency density meterincluding a vibrating section of a conduit supported by two orthogonalbias springs.

FIG. 22 illustrates another embodiment of the dual frequency densityincluding a straight vibrating section of a conduit supported by twoorthogonal bias springs.

FIG. 23 illustrates an embodiment of the dual frequency density meterincluding a pair of straight vibrating sections of a conduit disposed ina parallel arrangement.

FIG. 24 illustrates an embodiment of the dual frequency density meterincluding a pair of U-shaped sections of a conduit disposed in aparallel arrangement.

Operating Principles of the Invention

Newton's second law of motion governing the natural flexural vibrationstraight conduit disposed on the x-axis of a Cartesian coordinate systemcan be written in the following form: ##EQU1## where E is Young'smodulus of the material making up the wall of the conduit, I is themoment of intertia of the cross section of the conduit, y is the lateraldeflection of the conduit. A and B are constants, μ₀ and μ are theviscosity of the fluids occupying the exterior and interior spaces ofthe conduit, D is the internal diameter of the conduit, t is the time, mis the linear density of the conduit itself and ρ is the density of thefluid occupying the interior space of the conduit. It is readily foundthat equation (1) a solution of the following form: where X(x) standsfor a function of x, exp. stands for the exponential function and λ is acharacteristic value to be determined from the boundary conditions ofthe conduit. Substitution of equation (2) into (1) yields the equation##EQU2## The characteristic value λ is determined by applying theboundary condition the conduit to the solution of equation (3). Forexample, the numerical value of (λ^(1/4) L), where L is the length ofthe vibrating section of the conduit, for the primary mode of vibrationof the conduit determined theoretically is equal to 4.730 for a conduitwith both ends clamped or both ends free and 1.875 for a conduit withone end fixed and one end free.

According to the solution given by equation (2). the natural frequency fof the flexural vibration of the conduit is given by equation ##EQU3##

Equation (4) can be simplified for a vibrating conduit in air or vacuumwherein μ₀ is negligibly small as follows, ##EQU4## According toequation (4) or (5), the natural frequency of the flexural vibration ofa conduit is a function of the density and viscosity of fluid containedin the conduit. If a single section of a conduit is under two flexuralvibrations in two orthogonal lateral directions, or two sections of aconduit are two flexural vibrations at two different naturalfrequencies, the natural of the two flexural vibrations of a vibratingconduit in air or are given by equations ##EQU5## By subtractingequation (7) from equation (6), the term including the fluid iseliminated and a relationship relating the fluid density to the twonatural frequencies of the flexural vibration is obtained, ##EQU6##Equation (8) is a theoretical equation, which may be written in moregeneral form as follows : ##EQU7## where G and H are constants intrinsicto the solid mechanics property of the vibrating conduit, which areindependent of the property of the fluid contained in the conduit. Theconstants G and H are determined empirically by calibrating the dualfrequency vibrating tube density meter of the present invention by usingtwo different sample fluids of known density. Equation (9) is derivedbased on theory of mechanics. In real world, theory may deviate fromtrue reality up to certain extent. In place of equation (9), one may usean empirical counterpart of the form π=F(f₁,f₂) determined byexperiments instead of the theory. It has now been proven that thedensity of a fluid contained in a conduit can be determined from twodifferent natural frequencies of flexural vibrations of a singlevibrating section of a conduit experiencing two flexural vibrations intwo orthogonal lateral directions or from two different naturalfrequencies of flexural vibrations of two different sections of aconduit, as shown by equation (9) that is the basis of the operatingprinciples of the present invention.

DESCRIPTION OF THE ILLUSTRATED EMBODIMENTS

In FIG. 1 there is illustrated a perspective view of an embodiment ofdual frequency vibrating conduit density meter, that comprises a conduit1 including a 360 degree loop 2 intermediate two anchored port legs 3and 4 wherein the two generally straight sections 5 and 6 respectivelyincluded in the two halves of the conduit 1, extend towards each otherfrom the anchored extremities thereof and are connected to one anotherby the 360 degree loop section 2 of the conduit. The midsection 7 of thelooped conduit 1 is rigidly anchored to the frame 8. An electromagnet 9induces flexural vibrations on the two of the conduit. A pair of motionsensors 10 and 11 respectively detect flexural vibrations of the twohalves of the conduit 1. As shown in FIGS. 2 and 3, the two halves ofthe conduit 1 have different flexural stiffness and, consequently,vibrate at two different natural frequencies. It should be mentionedthat the two generally straight sections 5 and 6 of the conduit 1 may bedisposed in an arrangement parallel to one another and extendingrespectively from port legs in an over-hanging relationship wherein thetwo generally straight sections of the conduit are now connected to oneanother by a looped section of the conduit of loop angle generally equalto 540 degrees. Of course, the two generally straight sections 5 and 6of the conduit may be disposed in other arrangements wherein the angletherebetween is greater than zero degrees (in-line) and less than 180degrees (parallel), and the two generally straight sections areconnected to one another by a looped section of the conduit of loopangle in the range of 360 to 540 plus degrees.

In FIG. 2 there is illustrated a cross section of the first half of thelooped conduit 1 taken along plane 2--2 as shown in FIG. 1.

In FIG. 3 there is illustrated a cross section of the second half of thelooped conduit 1 taken along plane 3--3 as shown in FIG. 1. It isnoticed that the two halves of the looped conduit 1 have the sameinternal diameter and different external diameters. As a consequence,the two halves of the looped conduit 1 vibrate laterally at twodifferent natural frequencies. There are many other arrangements formaking the two laterally vibrating sections of the conduit vibrate attwo different natural frequencies, which may involve the two differentconduit cross sections of the two vibrating sections having differentmoments of inertia in the cross sections as exemplified by theillustrative embodiments shown in FIGS. 5 and 6, 13 and 15 or mayinclude external spring bias as exemplified by the illustrativeembodiments shown in FIGS. 16, 17, 18, 19, 21 and 22. Of course, assuggested by equations (3) and (4), two different natural frequencies ofthe two laterally vibrating sections of the conduit can be obtained byemploying different boundary conditions (rigidity or stiffness in theanchoring of the extremities of the vibrating sections of the conduit),while employing the identical cross section of the conduit for bothvibrating sections of the conduit.

The dual frequency vibrating conduit density meter of the presentinvention exemplified by the embodiment shown in FIGS. 1, 2 and 3 mayoperate in two different modes. In the first mode of operation, theelectromagnetic vibrator 9 imposes flexural vibrations on the two halvesof the conduit in an intermittent manner by applying a mechanicalimpulse thereto at a regular time interval, wherein the intermittentlyinduced flexural vibrations experience an attenuation in time before thenext impulse is applied by the electromagnetic vibrator. An electronicdata precessor detects the two different natural frequencies of theflexural vibrations from the signals supplied by the motion detectors 10and 11. The fluid density is determined by carrying out the algorithmdefined by equation (9) or empirical counterpart thereof, whichalgorithm is carried by the electronic data processor that is not shownin the illustrative embodiment. In the second mode of operation, theelectromagnet vibrator 9 applies a vibratory force in a frequency sweepmode, that is continuously repeated in time. The electronic dataprocessor determines the natural frequencies signals supplied by themotion detectors 10 and 11 by detecting the frequencies corresponding tothe maximum amplitudes in the signals generated in a frequency sweepmode. Once two different natural frequencies of the two vibratingsections are detected, they are substituted into equation (9) orempirical counterpart thereof to determine the fluid density, whichcalculation is performed by an electronic data Processor. In ordercompletely eliminate the effect of the ambient air resistance on theflexural vibrations of the conduit that is a source of error indetermining the fluid density, the vibrating sections of the conduit maybe sealed in an evacuated container. It should be mentioned that the twovibrating sections of the conduit may have different inner diametersand/or different outer diameters, if specific operating conditionsdemand such an arrangement, in which case B and D in equations (6) and(7) are no longer the common constants and, yet, these two equations canbe readily solved for the fluid density. All of the differentembodiments of the dual frequency vibrating conduit density meter shownas illustrative embodiment operate on essentially the same principles asthose described in conjunction with FIGS. 1, 2 and 3 apart from thespecific arrangements designed to obtain the two different naturalfrequencies of the flexural vibrations.

In FIG. 4 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter comprising a pair of U-shaped sections12 and 13 of the conduit disposed in series, wherein each of the twoU-shaped conduits includes an electromagnetic vibrator 14 and a motiondetector 15, which may be two separate and independent elements or anintegral element performing the two functions. The two U-shaped sections12 and 13 of the conduit have two different natural frequencies offlexural vibrations, wherein the vibratory motions generallyperpendicular to the plane including the U-shaped sections of theconduit, as the two U-shaped sections of the conduit have differentflexural stiffness.

In FIG. 5 there is illustrated a cross section of the first U-shapedsection 12 of the conduit taken along plane 5--5 as shown in FIG. 4.

In FIG. 6 there is illustrated a cross section of the second U-shapedsection 13 of the conduit taken along plane 6--6 as shown in FIG. 4. Asa design means providing two different natural frequencies of flexuralvibrations, the two side portions of the wall of the second U-shapedsection of the conduit are shaved off in order to reduce the moment ofinertia of the cross section about a plane including the U-shapedsection of the conduit.

In FIG. 7 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter comprising a U-shaped conduit 16 thatincludes a pair of generally parallel straight vibrating sections 17 and18 disposed in series. Each of the two straight vibrating sections 17and 18 of the conduit includes a motion detector 19, while a commonelectromagnetic vibrator 20 induces flexural vibrations on bothvibrating sections 17 and 18 in an intermittent or continuous manner asdescribed in conjunction with FIGS. 1, 2 and 3. Of course, the twogenerally straight vibrating sections 17 and 18 must have differentnatural frequencies of flexural vibration, which condition is realizedby employing different flexural stiffnesses of the conduits or differentboundary conditions for the two vibrating sections 17 and 18. It isevident that the two generally straight vibrating sections of theconduit may be disposed in an in-line arrangement similar to theembodiment shown in FIG. 20 instead of the parallel arrangement in theU-shaped conduit shown in FIG. 7.

In FIG. 8 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter including a pair of generally straightvibrating sections 21 and 22 of a conduit respectively extending fromthe two port legs in a parallel and over-hanging arrangement, whereinthe angled extremities 23 and 24 thereof are connected to a rigidsection 25 of conduit by a pair of flexible couplings 26 and 27. Theextremities of the vibrating sections 21 and 22 adjacent to the two portlegs as well as the rigid section 25 are anchored rigidly to a frame.The two vibrating sections 21 and 22 have different flexural stiffnessand, consequently, have two different natural frequencies of flexuralvibration, from which the density of the fluid is determined by themethod described in conjunction with equation (9). As another designimplementation, the pair of flexible couplings 26 and 27 may beinstalled respectively at the roots of the two straight sections of theU-shaped conduit adjacent to the two port legs, respectively.

In FIG. 9 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter comprising a pair of U-shaped vibratingsections 28 and 29 of a conduit arranged in series in terms of fluidflow and side-by-side in terms of geometrical arrangement, whichU-shaped vibrating sections are included in a generally oblong 540degree loop of the conduit disposed intermediate two port legs 30 and31. The two U-shaped vibrating sections have different naturalfrequencies of flexural vibrations. For the sake of simplicitycity ofthe illustration, the electromagnetic vibrator and the motion sensorsare not shown in FIG. 9 and other figures which follow.

In FIG. 10 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter including an over-hanging vibratingsection 32 of the conduit extending from the two port legs 33 and 34,which vibrating section 32 includes a pair of two straight sections 35and 36 respectively in a side-by-side arrangement, which vibratingsections are fused into a single vibrating cantilever beam and connectedto one another by generally 360 degree loop section 37 of the conduit.This over-hanging section 32 of the conduit vibrates laterally in twoorthogonal directions wherein the first direction of the flexuralvibration is parallel to the plane including both straight sections 35and 36 of the conduit and the second direction is perpendicular to thefirst direction. It is clear that the two flexural vibrations of theover-hanging vibrating section 32 of the conduit in the two differentdirections have different natural frequencies because of differentflexural stiffnesses thereof in the two directions. It should bementioned that the flexural vibrations of the over-hanging section 32 ofthe conduit in the two different directions may be induced by twodifferent electromagnetic vibrators respectively applying mechanicalimpulses or vibratory forces in the two different directions or a singleelectromagnetic vibrator applying mechanical impulse or vibratory forcein a direction intermediate the two directions of the flexuralvibrations. Of course, it is generally preferred to have two motiondetectors respectively detecting the flexural vibrations in the twodirections, even though a single motion detector disposed on a plane 45degrees to the two planes wherein the two flexural vibrations take placecan detect both flexural vibrations. It is evident that the straightportion of the over-hanging vibrating section 32 of the conduit can be asingle elongated member including two parallel flow passages thereininstead of the two sections 35 and 36 of the conduit tied or connectedto one another.

In FIG. 11 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter including a pair of U-shaped vibratingsections

38 and 39 of conduit included in a generally 360 degree oblong loop 40of conduit disposed intermediate two port legs 41 and 42, which U-shapedvibrating sections 38 and 39 have two different natural frequencies offlexural vibration

In FIG. 12 there is illustrated am embodiment of the dual frequencyvibrating conduit density meter comprising a single straight vibratingsection 43 of a conduit with two fixed ends respectively connected totwo port legs 44 and 45. The cross section of the straight vibratingsection 43 of the conduit has two different moments of inertia about twoorthogonal planes parallel to the central axis of the conduit and,consequently, two flexural vibrations of the straight vibrating section43 in two orthogonal lateral directions have two different naturalfrequencies.

In FIG. 13 there is illustrated a cross section of the straightvibrating section 43 of the conduit taken along plane 13--13 as shown inFIG. 12, which cross section shows the flow passage with a circularcross section and the conduit with an oblong circular cross section. Ofcourse, in place of the conduit having an oblong circular cross section,a conduit with a circular cross section having two sides shaved off asshown in FIG. 15 or other design such as a circular conduit with areinforcing fin welded along the conduit may be employed, as almost anynonaxisymmetric cross section of the conduit provides two flexuralvibrations in two orthogonal lateral directions occurring at twodifferent natural frequencies. The two flexural vibrations in the twoorthogolateral lateral directions may be induced by two electromagneticvibrators respectively installed on two orthogonal planes parallel tothe central axis of the conduit or by a single electromagnetic vibrator46 installed on a plane generally 45 degrees to both of the two planeswhereon the two flexural vibrations respectively take place.

In FIG. 14 there is illustrated another embodiment of the dual frequencyvibrating conduit density meter comprising a single straight vibratingsection 47 of a conduit disposed intermediate two port legs 48 and 49.One extremity of the vibrating section 47 is fixedly connected to thefirst port leg 48 rigidly anchored to a frame and the other extremity isconnected to the second port leg 49 rigidly anchored to the frame in alaterally flexible arrangement by means of the flexible coupling 50. Asthe two diagonally opposite sides of the external surface of thevibrating section 47 are shaved off, the two flexural vibrations in thetwo orthogonal lateral directions have two different naturalfrequencies.

In FIG. 15 there is illustrated a cross section of the vibrating section47 of the conduit taken along plane 15--15 as shown in FIG. 14. Aconduit with two shaved off sides as shown in the particular illustratedembodiment.

In FIG. 16 there is illustrated a further embodiment of the dualfrequency vibrating conduit density meter having essentially the sameconstruction as the embodiment shown in FIG. 14 with one exception. Thestraight vibrating section 52 of a conduit has an angled extremity 53connected to an exit port leg 54 by a flexible coupling 55, which angledextremity is supported by a compressive coil spring 56 disposedlaterally on a plane including the angled extremity 53 of the vibratingsection 52 of the conduit and the exit port leg 54. The vibratingsection 52 of the conduit with a circular or noncircular cross sectionvibrates in two orthogonal lateral directions respectively perpendicularand parallel to the plane including the central axis of the flexiblecoupling 55 and the coil spring 56 at two different natural frequencies.

In FIG. 17 there is illustrated yet another embodiment of the dualfrequency vibrating conduit density meter having essentially the sameconstruction as the embodiment shown in FIG. 16 with a few exceptions.One extremity of the vibrating section 57 of a conduit is rigidlyanchored to a frame, while the other extremity with a Tee coupling 58 isconnected to a Y-shaped port leg 59 by a pair of flexible couplings 60and 61. The straight vibrating section 57 of the conduit vibrate in twoorthogonal lateral directions respectively perpendicular and parallel toa plane including the central axis of the flexible couplings 60 and 61at two different natural frequencies.

In FIG. 18 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter comprising a single straight vibratingsection 62 of a conduit with two extremities respectively connected totwo port legs 63 and 64 by a pair of flexible couplings 65 and 66,respectively. One extremit the vibrating section 62 of the conduit issupported by a first leaf spring disposed on a first plane parallel tothe central axis of the conduit and anchored to a frame, while the otherextremity of the vibrating section 62 of the conduit is supported by asecond leaf spring 68 disposed on a second plane parallel to the centralaxis of the conduit and perpendicular to the first plane which secondleaf spring 68 is also anchored to the frame. One extremity of thevibrating section 62 of the conduit supported by the leaf spring 67vibrates in a first lateral direction perpendicular to the leaf spring67 at a first natural frequency, while the other extremity supported bythe leaf spring 68 vibrates in a second direction perpendicular to theleaf spring 68 at a second natural frequency.

In FIG. 19 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter comprising a pair of vibrating sections69 and 70 of a conduit respectively extending from two port legs 71 and72 towards each other in line and connected to one another by a flexiblecoupling 73. ONe extremity of the vibrating section 69 of the conduitadjacent to the flexiflexible flexible coupling 73 is reinforced by afirst leaf spring 74 disposed on a first plane parallel to the centralaxis of the conduit and anchored to a frame, which extremity vibrates ina first lateral direction perpendicular to the leaf spring 74 at a firstnatural frequency, while one extremity of the vibrating section 70 ofthe conduit adjacent to the flexible coupling 73 is reinforced by asecond leaf spring 75 disposed on a second plane parallel to the centralaxis of the conduit and perpendicular to the first plane and anchored tothe frame, which extremity vibrates in a second lateral directionperpendicular to the leaf spring 75 at a second natural frequency.

In FIG. 20 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter comprising a pair of straight vibratingsections 76 and 77 of the conduit disposed in series, wherein oneextremity of each of the two vibrating sections 76 and 77 includes aflexible coupling 78, while the other extremity is rigidly anchored to aframe. The two vibrating sections 76 and 77 of the conduit vibrate attwo different natural frequencies. It should be mentioned that theembodiment shown in FIG. 20 works equally well without the pair offlexible couplings respectively included in the two vibrating sections76 and 77 of the conduit.

In FIG. 21 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter comprising a generally straightvibrating section 79 of a conduit with two angled extremitiesrespectively connected to two port legs 80 and 81 by a pair of flexiblecouplings 82 and 83, respectively. The two extremities of the vibratingsection 79 of the conduit are reinforced forced by a first set of coilsprings 84, 85, etc. disposed on a first plane parallel to the centralaxis of the conduit and by a second set of coil springs 86, 87 etc.disposed on a second plane parallel to the central axis of the conduitand perpendicular to the first plane. The vibrating section 79 of theconduit vibrates transversely in two lateral directions respectivelyperpendicular to the first and second planes at two different naturalfrequencies. In this embodiment, the vibrating section 79 of the conduitexperiences little bending deflection, as the deflections allowing thetransverse vibrations are provided by the flexible couplings 82 and 83.

In FIG. 22 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter having essentially the same constructionas the embodiment shown in FIG. 21 with the only exception being thatthe vibrating section 88 of a conduit is connected to two port legs 89and 90 by a pair of flexible couplings 91 and 92 in an in-linearrangement instead of angled connections.

In FIG. 23 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter comprising a pair of vibrating sections93 and 94 of conduit disposed in a parallel arrangement and connected totwo common port legs 95 and 96 at two opposite extremities. The twovibrating sections 93 and 94 vibrate at two different naturalfrequencies. In place of the solid conduit sections 93 and 94 employedin the particular illustrative embodiment, conduit sections including aflexible coupling at one extremity thereof may be employed in theconstruction of the embodiment shown in FIG. 23.

In FIG. 24 there is illustrated an embodiment of the dual frequencyvibrating conduit density meter including a pair of U-shaped vibratingsections and 98 of conduit disposed in a parallel arrangement andconnected to two common port legs 99 and 100, which U-shaped vibratingsections 97 and 98 of the conduit vibrate at two different naturalfrequencies.

It should be mentioned that a dual frequency density meter employing oneor two solid vibrating conduit sections vibrating at high naturalfrequencies such as the embodiments shown in FIGS. 1, 4, 7, 9, 10, 11,12, 23 and 24 is the best choice for measuring density of a fluid thatis not subjected to cavitation or effervescence promoted by the highfrequency lateral vibrations of the vibrating sections of the conduit.If the fluid through the density meter is at a state wherein cavitationor effervescence take place, a dual frequency density meter employingone or two vibrating sections vibrating at low natural frequencies suchas the embodiments shown in FIGS. 8, 14, 16, 17, 18, 19, 20, 21, and 22should be employed in order to prevent the cavitation or effervescencepromoted by the lateral vibration of the flow passage, which can createa serious error in measuring the fluid density. It is generallyrecommended to seal the vibrating conduit sections of the dual frequencydensity meter within an evacuated container wherein only the inlet andoutlet flanges are disposed exterior to the evacuated container in orderto eliminate the clamping effect of the ambient air surrending thevibrating conduit sections, that can introduce an error in themeasurement of the fluid density.

While the principles of the present invention have now been made clearby the illustration embodiments, there will be many obviousmodifications of the structures, arrangements, proportions, elements andmaterials, which are particularly adapted to the specific workingenvironments and operating conditions in the practice of the inventionwithout departing from those principles. It is not desired to limit theinventions to the particular illustrative embodiments shown anddescribed and, accordingly, all suitable modifications and equivalentsmay be resorted to falling within the scope of the inventions as definedby the claims which follow.

What is claimed is:
 1. A method for measuring density of mediacomprising in combination:(a) creating two different natural frequenciesof flexural vibration of conduit containing the media by laterallyvibrating at least one conduit providing at least one flow passage; (b)measuring the two natural frequencies of flexural vibration; and (c)determining density of the media as a function of a ratio of a parameterto a differential combination of the squares of the two naturalfrequencies of flexural vibration, wherein effect of viscosity of themedia on the two natural frequencies of flexural vibration is eliminatedin determining the media density as said function of the two naturalfrequencies of flexural vibration.
 2. A method as set forth in claim 1wherein the two natural frequencies of flexural vibration are obtainedby laterally vibrating the conduit in two different lateral directions.3. A method as set forth in claim 2 wherein the two natural frequenciesare obtained by laterally vibrating the conduit in continuous mode.
 4. Amethod as set forth in claim 2 wherein the two natural frequencies areobtained by laterally vibrating the conduit in intermittent mode.
 5. Amethod as set forth in claim 2 wherein the two natural frequencies areobtained by laterally vibrating the conduit in frequency sweep mode,wherein natural frequency is determined from frequency corresponding tomaximum amplitude of flexural vibration in frequency domain.
 6. A methodas set forth in claim 1 wherein the two natural frequencies of flexuralvibration are obtained by laterally vibrating two different sections ofconduit connected to one another in series, wherein the two sections ofconduit respectively have the two different natural frequencies offlexural vibration.
 7. A method as set forth in claim 6 wherein the twonatural frequencies are obtained by laterally vibrating the two sectionsof conduit in continuous mode.
 8. A method as set forth in claim 6wherein the two natural frequencies are obtained by laterally vibratingthe two sections of conduit in intermittent mode.
 9. A method as setforth in claim 6 wherein the two natural frequencies are obtained bylaterally vibrating the two sections of conduit in frequency sweep mode,wherein natural frequency is determined from frequency corresponding tomaximum amplitude of flexural vibration in frequency domain.
 10. Amethod as set forth in claim 1 wherein the two natural frequencies offlexural vibration are obtained by laterally vibrating two conduitsproviding two parallel flow passages, wherein the two conduitsrespectively have the two different natural frequencies of flexuralvibration.
 11. A method as set forth in claim 10 wherein the two naturalfrequencies are obtained by laterally vibrating the two conduits incontinuous mode.
 12. A method as set forth in claim 10 wherein the twonatural frequencies are obtained by laterally vibrating the two conduitsin intermittent mode.
 13. A method as set forth in claim 10 wherein thetwo natural frequencies are obtained by laterally vibrating the twoconduits in frequency sweep mode, wherein natural frequency isdetermined from frequency corresponding to maximum amplitude of flexuralvibration in frequency domain.